# Super-expanders and warped cones

## TL;DR

This paper characterizes when families of graphs related to warped cones are expanders with respect to a Banach space, linking this property to spectral gaps in associated group representations, and provides examples of universal expanders.

## Contribution

It establishes a precise criterion connecting warped cone graphs being expanders with spectral gaps in group representations on Banach spaces, offering new universal expander examples.

## Key findings

- Graphs quasi-isometric to warped cone levels are expanders iff the representation has a spectral gap.
- Provides examples of graphs that are expanders for all Banach spaces of non-trivial type.
- Links geometric group theory with Banach space theory through spectral gap conditions.

## Abstract

For a Banach space $X$, we show that any family of graphs quasi-isometric to levels of a warped cone $\mathcal O_\Gamma Y$ is an expander with respect to $X$ if and only if the induced $\Gamma$-representation on $L^2(Y;X)$ has a spectral gap. This provides examples of graphs that are an expander with respect to all Banach spaces of non-trivial type.